There is free image hosting sites out there but they don’t offer easy or customizable options to add to a website. Example: 500px is free (up to a certain amount) but you can only grab a single image at a time and can’t embed a gallery.

I think the best options are to put your images up on any of the big free ones (Flickr, Facebook, Twitter, Instagram, Google Photos) and then use a service to customize and pull in your gallery or carousel. Some services are free some are paid.

*Note: Pinterest is not that good when it comes to uploading images and create galleries.

Flickr
Pros:

Free

Highly customizable

Responsive

Easy to use

Tons of features

Cons:

Only uses Flickr images

Imgur

Pros:

Free

Easy to paste a single image into the site

Cons:

No customization

Posting galleries is difficult

Instagram Only

Lightwidget

Pros:

Easy to use

Tons of features

Easy customizable

Responsive

Lazyload

The paid option can be cheap (each widget is a one-time payment of $10)